arXiv subjects

A. G. Ramm

Publications and source records attributed to A. G. Ramm.

At least 37 records · Page 2Linked to original sources

arXiv2015

A short proof of the existence of the solution to elliptic boundary problem

There are several methods for proving the existence of the solution to the elliptic boundary problem $Lu=f \text{\,\, in\,\,} D,\quad u|_S=0,\quad (*)$. Here $L$ is an elliptic operator of second order, $f$ is a given function, and uniqueness of the solution to problem (*) is assumed. The known methods for proving the existence of the solution to (*) include variational methods, integral equation methods, method of upper and lower solutions. In this paper a method based on functional analysis is proposed. This method is conceptually simple and technically is easy. It requires some known a priori estimates and a continuation in a parameter method.

math.AP↗

arXiv2015

Representation of vector fields

A simple proof is given for the explicit formula which allows one to recover a $C^2-$smooth vector field $A=A(x)$ in $\mathbb{R}^3$, decaying at infinity, from the knowledge of its $\nabla \times A$ and $\nabla \cdot A$. The representation of $A$ as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded $C^2-$smooth domain.

math.CA↗

arXiv2015

Integral equations and applications

The goal of this Section is to formulate some of the basic results on the theory of integral equations and mention some of its applications. The literature of this subject is very large. Proofs are not given due to the space restriction. The results are taken from the works mentioned in the references.

math.CA↗

arXiv2015

A symmetry result for strictly convex domains

Assume that $D\subset \R^2$ is a strictly convex domain with $C^2-$smooth boundary. {\bf Theorem.} {\em If $\int_De^{ix}y^ndxdy=0$ for all sufficiently large $n$, then $D$ is a disc.}

math.AP↗

arXiv2013

A solution to the Pompeiu problem

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of distributions, and $$\int_{σ(D)} f(x) dx = 0 \quad \forall σ\in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain, the closure $\bar{D}$ of which is diffeomorphic to a closed ball, and $S$ is its boundary. Then the comp$ is connected and path connected. By $G$ the group of all rigid motions of $\R^n$ is denoted. This group consists of all translations and rotations. A proof of the following theorem is given. Theorem 1. {\it Assume that $n=2$, $f\not\equiv 0$, and (*) holds. Then $D$ is a ball.} Corollary. {\it If the problem $(\nabla^2+k^2)u=0$ in $D$, $u_N|_S=0$, $u|_S=const\neq 0$ has a solution, then $D$ is a ball.} Here $N$ is the outer unit normal to $S$.

math.AP↗

arXiv2013

Asymptotics of a class of integrals

Consider an integral $I(s):=\int_0^T e^{-s(x^2-icx)}dx$, where $c>0$ and $T>0$ are arbitrary positive constants. It is proved that $I(s)\sim \frac{i}{sc}$ as $s\to +\infty$. The asymptotic behavior of the integral $J(s):=\int_0^Te^{s(x^2+icx)}dx$ is also derived. One has $J(s)\sim \frac{e^{sT^2+iscT}}{s(2T+ic)}$ as $s\to +\infty$.

math-ph↗

arXiv2013

Inverse scattering with the data at fixed energy and fixed incident direction

Consider the Schrödinger operator $-\nabla^2+q$ $ $q$, $q=q(x), x \in \mathbf{R}^3$. Let $A(β,α, k)$ be the corresponding scattering amplitude, $k^2$ be the energy, $α\in S^2$ be the incident direction, $β\in S^2$ be the direction of scattered wave, $S^2$ be the unit sphere in $\mathbf{R}^3$. Assume that $k=k_0 >0$ is fixed, and $α=α_0$ is fixed. Then the scattering data are $A(β)= A(β,α_0, k_0)=A_q(β)$ is a function on $S^2$. The following invers$ \textit{IP: Given an arbitrary $f \in L^2(S^2)$ and an arbitrary small number $$ $q \in C_0^{\infty}(D)$, where $D \in \mathbf{R}^3$ is an arbitrary fixed domai$ $||A_q(β)-f(β)||_{L^2(S^2)} < ε$?} A positive answer to this question is given. A method for constructing such a $q$ is proposed. There are infinitely many such $q$, not necessarily real-valued.

math-ph↗

arXiv2012

The Pompeiu problem

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}'$, where $\mathcal{S}'$ is the Schwartz class of distributions, and $$\int_{σ(D)} f(x) dx = 0 \quad \forall σ\in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain, the closure $\bar{D}$ of which is diffeomorphic to a closed ball. Then the complement of $\bar{D}$ is connected and path connected. Here $G$ denotes the group of all rigid motions in $\R^n$. This group consists of all translations and rotations. It is conjectured that if $f\neq 0$ and (*) holds, then $D$ is a ball. Other conjectures, equivalent to the above one, are formulated and discussed. Several new short proofs are given for the earlier proved results.

math.AP↗

arXiv2012

Large-time behavior of the weak solution to 3D Navier-Stokes equations

The weak solution to the Navier-Stokes equations in a bounded domain $D \subset \mathbb{R}^3$ with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all $t \geq 0$. In a bounded domain $D$ the solution decays exponentially fast as $t\to \infty$ if the force term decays at a suitable rate.

math-ph↗

arXiv2012

Scattering of electromagnetic waves by many nanowires

Electromagnetic wave scattering by many parallel to $z-$axis, thin, impedance, circular infinite cylinders is studied asymptotically as $a\to 0$. Let $D_m$ be the crossection of the $m-$th cylinder, $a$ be its radius, and $\hat{x}_m=(x_{m1},x_{m2})$ be its center, $1\le m \le M$, $M=M(a)$. It is assumed that the points $\hat{x}_m$ are distributed so that $$\mathcal{N}(Δ)=\frac 1{2πa}\int_ΔN(\hat{x})d\hat{x}[1+o(1)], $$ where $\mathcal{N}(Δ)$ is the number of points $\hat{x}_m$ in an arbitrary open subset $Δ$ of the plane $xoy$. The function $N(\hat{x})\geq 0$ is a given continuous function. An equation for the self-consistent (efficient) field is derived as $a\to 0$. A formula is derived for the effective refraction coefficient in the medium in which many thin impedance cylinders are distributed. These cylinders may model nanowires embedded in the medium. Our result shows how these cylinders influence the refraction coefficient of the medium.

math-ph↗

arXiv2012

Many-body wave scattering problems in the case of small scatterers

Formulas are derived for solutions of many-body wave scattering problems by small particles in the case of acoustically soft, hard, and impedance particles embedded in an inhomogeneous medium. The case of transmission (interface) boundary conditions is also studied in detail. The limiting case is considered, when the size $a$ of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived. The theory is based on a study of integral equations and asymptotics of their solutions as $a\to 0$. The case of wave scattering by many small particles embedded in an inhomogeneous medium is also studied.

math-ph↗

arXiv2012

Large-time behavior of solutions to evolution problems

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$, $H$ is a Hilbert space, $t\in \R_+:=[0,\infty)$, $A(t)$ is a linear dissipative operator: Re$(A(t)u,u)\le -γ(t)(u,u)$, %$γ(t)\ge 0$, $F(t,u)$ is a nonlinear operator, $|F(t,u)|\le c_0|u|^p$, $p>1$, $c_0,p$ are positive constants, $|b(t)|\le β(t),$ $β(t)\ge 0$ is a continuous function. Sufficient conditions are given for the solution $u(t)$ to problem (*) to exist for all $t\ge0$, to be bounded uniformly on $\R_+$, and a bound on $|u(t)|$ is given. This bound implies the relation $\lim_{t\to \infty}|u(t)|=0$ under suitable conditions on $γ(t)$ and $β(t)$.

math.CA↗

arXiv2012

Heat transfer in a medium in which many small particles are embedded

The heat equation is considered in the complex system consisting of many small bodies (particles) embedded in a given material. On the surfaces of the small bodies a Newton-type boundary condition is imposed. An equation for the limiting field is derived when the characteristic size $a$ of the small bodies tends to zero, their total number $\mathcal{N}(a)$ tends to infinity at a suitable rate, and the distance $d = d(a)$ between neighboring small bodies tends to zero $a << d$. No periodicity is assumed about the distribution of the small bodies.

math.AP↗

arXiv2012

Electromagnetic Wave Scattering by Small Impedance Particles of an Arbitrary Shape

Scattering of electromagnetic (EM) waves by one and many small ($ka\ll 1$) impedance particles $D_m$ of an arbitrary shape, embedded in a homogeneous medium, is studied. Analytic formula for the field, scattered by one particle, is derived. The scattered field is of the order $O(a^{2-κ})$, where $κ\in [0,1)$ is a number. This field is much larger than in the Rayleigh-type scattering. An equation is derived for the effective EM field scattered by many small impedance particles distributed in a bounded domain. Novel physical effects in this domain are described and discussed.

math.AP↗

arXiv2012

Inverse scattering problem for the Maxwell's equations

Inverse scattering problem is discussed for the Maxwell's equations. A reduction of the Maxwell's system to a new Fredholm second-kind integral equation with a {\it scalar weakly singular kernel} is given for electromagnetic (EM) wave scattering. This equation allows one to derive a formula for the scattering amplitude in which only a scalar function is present. If this function is small (an assumption that validates a Born-type approximation), then formulas for the solution to the inverse problem are obtained from the scattering data: the complex permittivity $\ep'(x)$ in a bounded region $D\subset \R^3$ is found from the scattering amplitude $A(β,α,k)$ known for a fixed $k=ω\sqrt{\ep_0 μ_0}>0$ and all $β,α\in S^2$, where $S^2$ is the unit sphere in $\R^3$, $\ep_0$ and $μ_0$ are constant permittivity and magnetic permeability in the exterior region $D'=\R^3 \setminus D$. The {\it novel points} in this paper include: i) A reduction of the inverse problem for {\it vector EM waves} to a {\it vector integral equation with scalar kernel} without any symmetry assumptions on the scatterer, ii) A derivation of the {\it scalar integral equation} of the first kind for solving the inverse scattering problem, and iii) Presenting formulas for solving this scalar integral equation. The problem of solving this integral equation is an ill-posed one. A method for a stable solution of this problem is given.

math.AP↗

arXiv2012

Spectral properties of Schrödinger-type operators and large-time behavior of the solutions to the corresponding wave equation

Let $L$ be a linear, closed, densely defined in a Hilbert space operator, not necessarily selfadjoint. Consider the corresponding wave equations &(1) \quad \ddot{w}+ Lw=0, \quad w(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. &(2) \quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0, where $k>0$ is a constant. Necessary and sufficient conditions are given for the operator $L$ not to have eigenvalues in the half-plane Re$z<0$ and not to have a positive eigenvalue at a given point $k_d^2 >0$. These conditions are given in terms of the large-time behavior of the solutions to problem (1) for generic $f$. Sufficient conditions are given for the validity of a version of the limiting amplitude principle for the operator $L$. A relation between the limiting amplitude principle and the limiting absorption principle is established.

math.AP↗

arXiv2012

Convergence of Time-Dependent Turing Structures

Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the $L^2$ norm of the solution. This method is appropriate for the equations with time dependent coefficients. It yields new results and is applicable when the usual linearization method is not applicable.

math.FA↗

arXiv2012

Dynamical Systems Method for solving nonlinear operator equations in Banach spaces

Let $F(u)=h$ be a solvable operator equation in a Banach space $X$ with a Gateaux differentiable norm. Under minimal smoothness assumptions on $F$, sufficient conditions are given for the validity of the Dynamical Systems Method (DSM) for solving the above operator equation. It is proved that the DSM (Dynamical Systems Method) \bee \dot{u}(t)=-A^{-1}_{a(t)}(u(t))[F(u(t))+a(t)u(t)-f],\quad u(0)=u_0,\ %\dot{u}=\frac{d u}{dt}, \eee converges to $y$ as $t\to +\infty$, for $a(t)$ properly chosen. Here $F(y)=f$, and $\dot{u}$ denotes the time derivative.

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